## Abstract

This paper examines a parameterized problem that we refer to as n - k GRAPH COLORING, i.e., the problem of determining whether a graph G with n vertices can be colored using n - k colors. As the main result of this paper, we show that there exists a O(kn^{2} + k^{2} + 2^{3.8161k}) = O(n^{2}) algorithm for n - k GRAPH COLORING for each fixed k. The core technique behind this new parameterized algorithm is kernalization via maximum (and certain maximal) matchings. The core technical content of this paper is a near linear-time kernelization algorithm for n - k CLIQUE COVERING. The near linear-time kernelization algorithm that we present for n - k CLIQUE COVERING produces a linear size (3k - 3) kernel in O(k(n + m)) steps on graphs with n vertices and m edges. The algorithm takes an instance (G, k) of CLIQUE COVERING that asks whether a graph G can be covered using |V| - k cliques and reduces it to the problem of determining whether a graph G′=(V′,E′) of size ≤ 3k - 3 can be covered using |V′| - k′ cliques. We also present a similar near linear-time algorithm that produces a 3k kernel for VERTEX COVER. This second kernelization algorithm is the crown reduction rule.

Original language | English |
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Title of host publication | Graph-Theoretic Concepts in Computer Science |

Subtitle of host publication | 30th International Workshop, WG 2004, Bad Honnef, Germany, June 21-23, 2004. Revised Papers |

Editors | Juraj Hromkovič, Manfred Nagl , Bernhard Westfechtel |

Publisher | Springer Berlin Heidelberg |

Pages | 257-269 |

Number of pages | 13 |

Volume | 3353 |

ISBN (Electronic) | 978-3-540-30559-0 |

ISBN (Print) | 978-3-540-24132-4 |

DOIs | |

State | Published - 2004 |

### Publication series

Name | Lecture Notes in Computer Science |
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Publisher | Springer Verlag |

ISSN (Print) | 0302-9743 |

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